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Theorem 3sstr3i 3137
 Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr3.1 𝐴𝐵
3sstr3.2 𝐴 = 𝐶
3sstr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3sstr3i 𝐶𝐷

Proof of Theorem 3sstr3i
StepHypRef Expression
1 3sstr3.1 . 2 𝐴𝐵
2 3sstr3.2 . . 3 𝐴 = 𝐶
3 3sstr3.3 . . 3 𝐵 = 𝐷
42, 3sseq12i 3125 . 2 (𝐴𝐵𝐶𝐷)
51, 4mpbi 144 1 𝐶𝐷
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ⊆ wss 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084 This theorem is referenced by: (None)
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